Custom Post Types, Custom Taxonomies and Permalinks ?

You need to write a custom rule. I did similar stuff for a recent project. Create a template, and assign it to a page, with slug as ‘clients’ (or anything you like, but change the slug in the following rule as well). Try the following code, use functions.php:

add_action('init', 'custom_rule');
function custom_rule()
{
    global $wp, $wp_rewrite;
    $wp_rewrite->add_rule('^projects/clients/', 'index.php?pagename=clients', 'top');
}

Then on that template, form a custom query with your desired params (clients taxonomy) and loop through it.

Note: After you put the above code in your functions.php file, go to
Settings > Permalinks, and click on save.

Hope this helps (I recommend you try it locally first)!